Page 548 - Physics Coursebook 2015 (A level)

Cambridge International A Level Physics
Remember – you always add uncertainties; never subtract. Where quantities are:
■■ added or subtracted, then add absolute uncertainties
■■ multiplied or divided, then add percentage uncertainties.
WORKED EXAMPLE
536
QUESTIONS
6 In the expressions that follow, x and y are variables in an experiment. All the other quantities in the expressions are constants.
In each case, state the graph you would plot to produce a straight line. Give the gradient of each line in terms of the constants in the expression.
When quantities are added or subtracted, their absolute uncertainties are added. A simple example is measuring the length of a stick using a millimetre scale. There is likely to be an uncertainty of 0.5 mm at both ends, giving a total uncertainty of 1.0 mm.
When quantities are multiplied or divided, combining uncertainties is a little more complex. To find the combined uncertainty in this case, we add the percentage uncertainties of the two quantities to find the total percentage uncertainty.
a y=kx3/2 b y=cxq
c m=8x By2
d y=y0ekx
e R=(y−y0)
x2
7 The period of oscillation T of a small spherical mass supported by a length l of thread is given by the expression:
T=2π gl
where g is the acceleration due to gravity.
Design a laboratory experiment using this expression to determine the acceleration due to gravity. You should draw a diagram showing the arrangement of your equipment. In your account, you should pay particular attention to:
a the procedure to be followed
b the measurements to be taken
c analysis of the data to determine g
d any safety precautions that you would take.
Treatment of uncertainties
All results should include an estimate of the absolute uncertainty. For example, when measuring the time for a runner to complete the 100 m you may express this as 12.1 ± 0.2 s. This can also be expressed as a percentage uncertainty (see Chapter P1); the percentage uncertainty
is equal to 0.2 × 100% = 1.65%, so we write the value as 12.1
12.1s±1.7%, or even 12.1s±2%.
Combining uncertainties
When quantities are combined, what is the uncertainty in the result?
Suppose that quantity A = 1.0 ± 0.1 and that
B = 2.0 ± 0.2, so that the value of A + B is 3.0. The maximum likely value of A + B, taking into account the uncertainties, is 3.3, and the minimum likely value is 2.7. You can see that the combined uncertainty is ± 0.3, so A+B = 3.0±0.3. Similarly B−A = 1.0±0.3.
2 The potential difference across a resistor is measured as 6.0 ± 0.2 V, while the current is measured as 2.4 ± 0.1 A.
Calculate the resistance of the resistor and the absolute uncertainty in its measurement.
Step1 Findthepercentageuncertaintyineachof the quantities:
percentage uncertainty in p.d. = 0.2 × 100% = 3.3% 6.0
percentage uncertainty in current = 0.1 × 100% 2.4
= 4.2% Step2 Addthepercentageuncertainties.
sum of uncertainties = (3.3 + 4.2)% = 7.5%
Step3 Calculatetheresistancevalueandfindthe absolute uncertainty
R= V = 6.0 =2.5Ω I 2.4
7.5%of2.5=0.1875≈0.2Ω
The resistance of the resistor is 2.5 ± 0.2 Ω.
When you calculate the uncertainty in the square of a quantity, since this is an example of multiplication, you should double the percentage uncertainty. For example, if A = 2.0 ± 0.2 cm, then A has a percentage uncertainty of 10% so A2 = 4.0 m2 ± 20%; or giving the absolute uncertainty, A2 = 4.0±0.8cm2.